Journal of Applied Probability

Estimation of integrals with respect to infinite measures using regenerative sequences

Krishna B. Athreya and Vivekananda Roy

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Abstract

Let f be an integrable function on an infinite measure space (S, S, π). We show that if a regenerative sequence {Xn}n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on Rd using a simple symmetric random walk on Z.

Article information

Source
J. Appl. Probab., Volume 52, Number 4 (2015), 1133-1145.

Dates
First available in Project Euclid: 22 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1450802757

Digital Object Identifier
doi:10.1239/jap/1450802757

Mathematical Reviews number (MathSciNet)
MR3439176

Zentralblatt MATH identifier
1334.65003

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 60F05: Central limit and other weak theorems

Keywords
Markov chain Monte Carlo improper target random walk regenerative sequence

Citation

Athreya, Krishna B.; Roy, Vivekananda. Estimation of integrals with respect to infinite measures using regenerative sequences. J. Appl. Probab. 52 (2015), no. 4, 1133--1145. doi:10.1239/jap/1450802757. https://projecteuclid.org/euclid.jap/1450802757


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References

  • Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.
  • Athreya, K. B. and Roy, V. (2014). Monte Carlo methods for improper target distributions. Electron. J. Statist. 8, 2664–2692.
  • Baron, M. and Rukhin, A. L. (1999). Distribution of the number of visits of a random walk. Commun. Statist. Stoch. Models 15, 593–597.
  • Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.
  • Casella, G. and George, E. I. (1992). Explaining the Gibbs sampler. Amer. Statistician 46, 167–174.
  • Crane, M. A. and Iglehart, D. L. (1975). Simulating stable stochastic systems. III. Regenerative processes and discrete-event simulations. Operat. Res. 23, 33–45.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
  • Glynn, P. W. and Iglehart, D. L. (1987). A joint central limit theorem for the sample mean and regenerative variance estimator. Ann. Operat. Res. 8, 41–55.
  • Hobert, J. P., Jones, G. L., Presnell, B. and Rosenthal, J. S. (2002). On the applicability of regenerative simulation in Markov chain Monte Carlo. Biometrika 89, 731–743.
  • Karlsen, H. A. and Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Ann. Statist. 29, 372–416.
  • Kasahara, Y. (1984). Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions. J. Math. Kyoto Univ. 24, 521–538.
  • Metropolis, N. et al. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Mykland, P., Tierney, L. and Yu, B. (1995). Regeneration in Markov chain samplers. J. Amer. Statist. Assoc. 90, 233–241.
  • Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd edn. Springer, New York.
  • R Development Core Team (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing Vienna, Austria. Available at http://www.r-project.org.
  • Skorokhod, A. V. (1957). Limit theorems for stochastic processes with independent increments. Theory Prob. Appl. 2, 138–171.